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Progress to Advanced Mathematics

A book designed to aid progression towards AS Level, A Level, the IB and other advanced mathematics examinations.

Progress to Advanced Mathematics

Written by a team of highly experienced mathematics teachers, this book provides carefully selected materials to aid progress towards AS Level, A Level, the IB and other advanced mathematics examinations.

Students will find Progress to Advanced Mathematics helps to build their confidence by ensuring they have a good grounding in core material before starting advanced topics.

In particular, the book will help to develop the fluency in algebra necessary for success in later work.

Progress to Advanced Mathematics provides a series of exercises which can be used either before embarking on sixth form studies or throughout the teaching of the course — as introductory material before each topic — and will help students to:

• practise the essential skills needed for AS and A Level;
• consolidate material from GCSE;
• develop proficiency in algebra;
• gain a firm understanding of the basics of graphs, coordinate geometry and trigonometry.

The book has over seventy exercises of carefully graded questions. Answers are provided for all the questions and there is a two-page index.

Each chapter ends with a miscellaneous exercise, which may be used either for consolidation or to help identify areas where further practice is needed.

Sample pages can be seen in the Book Gallery.

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• Order form (PDF)

Learning outcomes

A Checklist of learning outcomes for pupil use is available as a free PDF document. Teachers may freely print or photocopy the checklist for distribution to students. The document may be viewed or downloaded using the link:

• Checklist of learning outcomes (PDF)

Preface

We have written this book in response to the experience of teachers who are familiar with the difficulties students face as they make the transition from GCSE, or equivalent, to AS Level, A Level and other advanced mathematics examinations. Progress to Advanced Mathematics aims to make the transition smoother and more motivating. We designed the sister book Progress to Advanced Mathematics with a similar purpose for the Scottish system, and we have been encouraged by the reception it has received.

Students will find this book helps to build their confidence by ensuring they have a good grounding in fundamental material before starting advanced topics. In particular, the book will help to develop the fluency in algebra necessary for success in later work.

Progress to Advanced Mathematics provides a series of exercises which can be used either before embarking on sixth form studies or throughout the teaching of the course as introductory material before each topic. Proficiency in algebra is emphasised, but the basics of graphs, coordinate geometry and trigonometry are also covered.

Sometimes a technique studied earlier is needed in a slightly different form in advanced work. For example, though GCSE students will have solved simple linear equations, they may not have solved equations that have decimal coefficients. In order to provide practice in such techniques, we have included questions of an appropriate type in the exercises.

We have designed the material to suit students with any starting level of competence, from basic to well beyond GCSE. The range of exercises should enable students of all abilities to practise the techniques essential to improving their performance in an advanced course. Each section has clearly defined learning outcomes, allowing exercises to be used individually or as part of a complete unit.

Within each exercise the questions are carefully graded. In addition, foundation questions are indicated by an anchor symbol and extension questions are indicated by a snowflake symbol. Each chapter ends with a miscellaneous exercise, to provide consolidation as well as helping to identify areas where further practice is needed.

Contents

Progress to Advanced Mathematics contains over 70 exercises organised into nine chapters, as shown below.

Answers are given to all the exercises and there is a two-page index.

1. Basic algebra
   1. Simplifying expressions
   2. Expanding brackets
   3. Factorising by finding a common factor
   4. Factorising quadratic expressions
   5. Substituting into expressions
   6. Miscellaneous questions
2. Linear equations
   1. Simple linear equations
   2. Linear equations with fractions
   3. Linear inequalities
   4. Simultaneous linear equations
   5. Miscellaneous questions
3. Coordinates
   1. Coordinates
   2. The midpoint of a line segment
   3. The length of a line segment
   4. Straight lines parallel to the axes
   5. Areas
   6. Circles
   7. Miscellaneous questions
4. Triangles
   1. Sine, cosine and tangent
   2. The sine rule
   3. The cosine rule
   4. The area of a triangle
   5. Miscellaneous questions
5. Straight lines
   1. The gradient of a straight line
   2. The equation of a straight line
   3. Parallel lines
   4. Perpendicular lines
   5. The intersection of two lines
   6. Miscellaneous questions
6. Quadratic equations
   1. Quadratic equations by factorising
   2. Using the quadratic formula
   3. Special quadratic equations
   4. Simultaneous equations where one is non-linear
   5. Miscellaneous questions
7. Curves
   1. Simple quadratic curves
   2. Quadratic curves in factorised form
   3. Cubic curves
   4. The intersection of a line and a quadratic curve
   5. The intersection of two quadratic curves
   6. The intersection of a line and a curve
   7. Miscellaneous questions
8. Fractions, indices and surds
   1. Indices
   2. Equations with powers
   3. Algebraic fractions
   4. Equations with algebraic fractions
   5. Surds
   6. Miscellaneous questions
9. More algebra
   1. Constructing equations
   2. Changing the subject
   3. Completing the square
   4. Functions
   5. Miscellaneous questions

© A K Jobbings 2004–2012